Related papers: Optimal Reach Estimation and Metric Learning
Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory.…
Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great…
We propose a robust and scalable procedure for general optimization and inference problems on manifolds leveraging the classical idea of `median-of-means' estimation. This is motivated by ubiquitous examples and applications in modern data…
We study the problem of robustly estimating the mean of a $d$-dimensional distribution given $N$ examples, where most coordinates of every example may be missing and $\varepsilon N$ examples may be arbitrarily corrupted. Assuming each…
Measuring geometric similarity between high-dimensional network representations is a topic of longstanding interest to neuroscience and deep learning. Although many methods have been proposed, only a few works have rigorously analyzed their…
We study the properties of nonparametric least squares regression using deep neural networks. We derive non-asymptotic upper bounds for the prediction error of the empirical risk minimizer of feedforward deep neural regression. Our error…
This paper deals with a nonparametric shape respecting estimation method for U-shaped or unimodal functions. A general upper bound for the nonasymptotic L_1-risk of the estimator is given. The method is applied to the shape respecting…
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…
In machine learning, accurately predicting the probability that a specific input is correct is crucial for risk management. This process, known as uncertainty (or confidence) estimation, is particularly important in mission-critical…
Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local…
The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on…
Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…
Accurate approximation of probability measures is essential in numerical applications. This paper explores the quantization of probability measures using the maximum mean discrepancy (MMD) distance as a guiding metric. We first investigate…
We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in R^D given a noisy sample from the manifold. We assume that the manifold satisfies a smoothness condition and that the noise…
We study nonparametric distance-based (isotropic) local polynomial methods for estimating the boundary average treatment effect curve, a causal functional that captures treatment effect heterogeneity in boundary discontinuity designs. We…
In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators under strong absorption conditions. We establish improved geometric regularity along the free boundary, for a sharp value…
Optimal estimation is a promising tool for estimation of payloads' inertial parameters and localization of robots in the presence of multiple contacts. To harness its advantages in robotics, it is crucial to solve these large and…
The existing approaches to intrinsic dimension estimation usually are not reliable when the data are nonlinearly embedded in the high dimensional space. In this work, we show that the explicit accounting to geometric properties of unknown…
We derive fundamental asymptotic results for the expected covering radius $\rho(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For…
We study deformations of the geodesic distances on a domain of R N induced by a function called conformal factor. We show that under a positive reach assumption on the domain (not necessarily a submanifold) and mild assumptions on the…